In the first photo, the initial pattern (the graph) is started correctly with the two purl bumps in white/green toward the center of the scarf and it looks like it continues correctly. The background does appear to be correct in this picture area. But the initial rows of the scarf appear reversed and in the second photo you can see the initial rows of the scarf which are all background, have an error. You shouldn't be seeing white purl bumps in the valleys that are done in stockinette (white). These are the directions to start the scarf:
Directions:
CO 31st in black. K 2 rows in black.
Background Section:
Row 1: K across in green
Row 2: P across in green
Row 3: K across in black
Row 4: K across in black

Make sure you cast on in black and then K 2 rows in black. The next row is K white/green and the following is P white/green. These two rows will give you stockinette (V shaped sts in the contrast color on the RS) and you won't see purl bumps in the contrast color background.
When you start chart 1 (white/green), make susre that the side with the contrast color valleys is facing you for chart row 1. You'll see the first 2 purl bumps of the snake appear in row 2 and the purl bumps will be on the RS.

The "trick" to this pattern is that since it's an "illusion" you can see it better from an angle, AND after you've completely quite a few repeats.
I was having a similar problem in the beginning, in fact I almost gave up because it looked so WRONG to me.
I put it down for a few days and came back to it.
After it was done, I liked it so much, I made a second one.

Don't give up, if you're following the pattern/chart it should start looking like the pic soon.

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So it is hard to see, but I did count out 31st in my white which the pattern says black. Then, using my white/black, I did K two rows. Switching to the green, I K and then P two rows of green to make a valley between the green. I see what's wrong now, but how do I keep it from happening again on the other end of the chart? Do I need to switch something from the directions? Or am I just really bad at this??

So it is hard to see, but I did count out 31st in my white which the pattern says black. Then, using my white/black, I did K two rows. Switching to the green, I K and then P two rows of green to make a valley between the green. I see what's wrong now, but how do I keep it from happening again on the other end of the chart? Do I need to switch something from the directions? Or am I just really bad at this??

Aha, this may be part at least of the problem. The directions say to P one row, not 2. Purling 2 rows is giving you the line of bumps in white that shows up in photo 2 in the first and second white stripes after the cast on. Your knitting looks beautiful and this scarf will work out really well once you get past the initial rows and onto the chart.
You have the background sts correctly done once you get to the picture part of the pattern, where you're following the chart (look at those nice white stockinette valleys in the upper part of photo 1). So why not follow the pattern set up by the first say 5sts of the chart (background area), when you do the first 14 rows of the scarf? Get that background right and then start the chart making sure that you begin the chart with the RS (stockinette valleys) facing you.